Improved phase-change memory cells are provided for storing information in a plurality of programmable cell states. A phase-change material is located between first and second electrodes for applying a read voltage to the phase-change material to read the programmed cell state. An electrically-conductive component extends from one electrode to the other in contact with the phase-change material. The resistance presented by this component to a cell current produced by the read voltage is less than that of the amorphous phase and greater than that of the crystalline phase of the phase-change material in any of the cell states.